Earth and moon gravitational ratios and proportionalities

Question

This question is for an astronomy class of mine. I don't believe the question is particularly difficult, but rather I'm having trouble understanding what it wants me to do. We haven't covered ratios and proportionalities and as a result I'm very lost. Here's the question:

The gravitational acceleration due to a planet or moon tells you how “heavy” you feel when you stand on the planet or moon. Gravitational acceleration is proportional to the mass and radius of the planet or moon in the following way: $g\propto \frac{M}{R^2}$

1. Start by writing two proportionalities, one for the Earth, one for the Moon. Use the following variables:

• $M_{E}$ : mass of Earth
• $M_{M}$ : mass of Moon
• $R_{E}$ : radius of Earth
• $R_{M}$ : radius of Moon
• $g_{E}$ : gravitational acceleration of the Earth
• $g_{M}$ : gravitational acceleration of the Moon

2. Following the example under the ratio section, divide the two proportionalities to form a ratio, i.e., the ratio between the gravitational acceleration of Earth and the gravitational acceleration of the Moon. In other words, find the following in equation form: $$\frac{g_{E}}{g_{M}}$$

This isn't from a lack of effort. I'm just not entirely sure how to write or divide a proportionality.

Answer 1

The first part is just replacing $M$ with $M_E$ and so forth. You get two statements of proportionality, one with the $E$ values and one with the $M$ values.

The second part is algebraic division. Mathematically if $a\propto b$, and if $a_1,a_2,b_1,b_2$ are specific values that are in this proportion. $a_1=kb_1$ and $a_2=kb_2$ for some constant $k$. Division gives $a_1/a_2 = b_1/b_2$. The constants cancel

By division, you turn a statement of proportionally into an equation.

Answer 2

Saying $g\propto \frac{M}{R}$ is the same as saying that there exists a constant $k$ for which $g = k\frac{M}{R}$. This constant $k$ is the same for every planet or moon.

In these equations, you can replace $M$ and $R$ by the mass and ratio of the Earth to get the proportionality for the Earth. Then you can replace $M$ and $R$ with the mass and ratio of any other moon or planet and get the proportionality for that object.

In the 2nd part, you must obtain

the ratio between the gravitational acceleration of Earth and the gravitational acceleration of the Moon

Here, you just have to write $ \frac{g_E}{g_M} $ and replace $g_E$ and $g_M$ with the expressions you determined in the first question.

$\frac{g_E}{g_M} = \frac{k\frac{M_E}{R_E}}{k\frac{M_M}{R_M}}$

As you can see, the $k$ cancels out in the numerator and the denominator and leaves you with an expression of $\frac{g_E}{g_M}$ as a function of $M_E$, $M_M$, $R_E$ and $R_M$.